On Jacobi's transformation theory of elliptic functions

The main interpretative challenge set by the Fundamenta Nova Theoriae Functionum Ellipticarum lies in Jacobi's transformation theory upon which the entire theoretical edifice of the treatise depends. Unfortunately, Jacobi did not convey any indication of how he attained his general formulae for rational transformations of elliptic functions. He limited himself to providing a posteriori verification of the validity of his claims. The aim of this paper is precisely to describe the heuristic path by which in 1827 Jacobi succeeded in finding these transformation formulae. The proposed historical reconstruction will hopefully shed new light upon the emergence in Jacobi's work of the inversion process of elliptic integrals of the first kind and thus of the elliptic function sinamu itself.